Weierstrass's Monster
Weierstrass’s nowhere-differentiable but everywhere-continuous function prompts readers to share implementations, visualize it as audio, and connect it to a broader family of “pathological” examples like the Dirichlet, Thomae, Cantor and Conway base‑13 functions that break everyday calculus intuition. Many comments explore how such counterexamples drove the push for rigor in real analysis and topology, inspired classic books of counterexamples, and underpin modern ideas like Brownian motion and fractal interpolation. Others question the practical value of non-computable reals and exotic constructions, sparking side debates on computability, different sizes of infinity, and what it really means for a mathematical object to “exist.”
Implementation, Code, and Audio Experiments
- Commenters share simple implementations of Weierstrass-type series in Python and TypeScript.
- The function’s graph looks like an audio waveform; people experiment with sonifying it and share YouTube examples.
- Observations: audio only reflects finitely many series terms (band-limited spectrum); one can swap the “audible” part with another signal (e.g., Chopin) while keeping a nowhere-differentiable function mathematically.
Counterexamples and Pedagogy
- Multiple “counterexamples in analysis/topology” books are recommended; Weierstrass appears prominently alongside many related constructions.
- Some praise these examples as essential for understanding why theorem hypotheses matter; others feel it can start to feel like “cheating.”
- There is discussion of rigorous proof culture, especially in French education, and how such counterexamples historically pushed math toward greater rigor.
Other Pathological Functions and Constructions
- Frequently mentioned: Dirichlet function, its “continuous only at 0” variant, Thomae’s (popcorn) function, Cantor function (Devil’s staircase), indicator functions of Cantor sets, Conway base-13 function, discrete metric, Schwarz lantern, staircase paradox, Gabriel’s horn.
- Several people note that “most” continuous functions are nowhere differentiable (via Baire category), making Weierstrass typical rather than exotic.
Computability, Measure, and Foundations
- One view: because non-computable reals dominate (full measure, uncountable), most wild examples are “nonexistent” computationally and of dubious practical value.
- Corrections: computable reals form a countable, measure-zero subset; all reals used in practice so far are computable.
- Long subthread debates Cantor’s diagonal argument, different infinities, density of rationals, separability, and whether future mathematics might reject some current axioms.
- Constructivist perspectives appear, questioning classical cardinality arguments but noting that Weierstrass itself is constructively well-behaved.
Intuition, Limits, and Geometry Paradoxes
- Several users discuss intuition-breaking examples: the staircase paradox, shapes with fixed perimeter but varying area, convergence modes (pointwise vs uniform vs weak*), and non-continuity of “perimeter” under limits.
Applications and Miscellany
- Brownian motion and related stochastic models (e.g., Langevin dynamics) are cited as real-world uses of nowhere-differentiable paths.
- Mentions of fractal interpolation functions, functions with first but not second derivatives, and questions about integrability of “random” functions and the role of the axiom of choice.